[Paper Review] Graph Convolutional Networks for Simulating Multi-phase Flow and Transport in Porous Media

Jiamin Jiang, Bo Guo|arXiv (Cornell University)|Jul 10, 2023

Lattice Boltzmann Simulation Studies4 citations

TL;DR

This paper proposes a Graph Convolutional Network (GCN)-based surrogate model to simulate multi-phase flow and transport in porous media, addressing the limitations of convolutional neural networks (CNNs) on unstructured meshes. By incorporating an EdgeConv-based architecture tailored for hyperbolic PDE dynamics, the model achieves high-accuracy, stable predictions across multiple timesteps and generalizes effectively to unseen irregular and unstructured meshes, including PEBI and irregular Cartesian grids.

ABSTRACT

Numerical simulation of multi-phase fluid dynamics in porous media is critical for many energy and environmental applications in Earth's subsurface. Data-driven surrogate modeling provides computationally inexpensive alternatives to high-fidelity numerical simulators. While the commonly used convolutional neural networks (CNNs) are powerful in approximating partial differential equation solutions, it remains challenging for CNNs to handle irregular and unstructured simulation meshes. However, simulation models for Earth's subsurface often involve unstructured meshes with complex mesh geometries, which limits the application of CNNs. To address this challenge, we construct surrogate models based on Graph Convolutional Networks (GCNs) to approximate the spatial-temporal solutions of multi-phase flow and transport processes in porous media. We propose a new GCN architecture suited to the hyperbolic character of the coupled PDE system, to better capture transport dynamics. Results of 2D heterogeneous test cases show that our surrogates predict the evolutions of pressure and saturation states with high accuracy, and the predicted rollouts remain stable for multiple timesteps. Moreover, the GCN-based models generalize well to irregular domain geometries and unstructured meshes that are unseen in the training dataset.

Motivation & Objective

• Address the challenge of applying deep learning to multi-phase flow simulations in porous media with complex, unstructured meshes.
• Overcome the limitations of CNNs, which are restricted to regular, structured grids and struggle with irregular mesh geometries.
• Develop a GCN-based surrogate model that captures the hyperbolic nature of saturation dynamics in coupled PDE systems.
• Ensure generalization to unseen domain geometries and unstructured meshes, such as PEBI and irregular Cartesian grids.
• Enable fast, accurate, and stable simulation rollouts for uncertainty quantification in subsurface applications.

Proposed method

• Represent the simulation domain as a graph, with nodes as mesh cells and edges as connectivity between adjacent cells.
• Design a GCN architecture that integrates GraphConv layers with EdgeConv to better capture local spatial and dynamic variations in saturation fields.
• Train the model using supervised learning on per-node pressure and saturation states from high-fidelity numerical simulations.
• Use message passing mechanisms to aggregate neighborhood information across multiple GCN layers, enabling learning of complex physical relationships.
• Incorporate max pooling and residual connections to stabilize training and improve feature representation.
• Fine-tune the model on additional PEBI mesh samples to improve generalization to unstructured grids not seen during initial training.

Experimental results

Research questions

• RQ1 Can a GCN-based surrogate model accurately predict the spatial-temporal evolution of pressure and saturation in multi-phase flow through heterogeneous porous media?
• RQ2 How does the proposed GCN architecture with EdgeConv compare to standard GCN or CNN models in capturing sharp saturation fronts and discontinuities?
• RQ3 To what extent can the trained GCN model generalize to unseen domain geometries and unstructured meshes such as PEBI and irregular Cartesian grids?
• RQ4 Does the surrogate model maintain prediction stability over long simulation rollouts despite being trained only on next-step predictions?
• RQ5 Can the GCN surrogate achieve significant computational speedups while preserving accuracy compared to high-fidelity numerical simulators?

Key findings

• The GCN-based surrogate model achieves mean relative saturation errors of 0.91%, 0.85%, and 0.93% across three test cases on regular Cartesian meshes, significantly outperforming a standard GraphConv-only model with errors of 1.89%, 1.68%, and 1.32%.
• The model with EdgeConv preserves fine-scale heterogeneities and sharp saturation fronts, avoiding the smearing effect observed in models without EdgeConv.
• The surrogate model generalizes well to irregular Cartesian meshes and PEBI unstructured meshes, with minimal prediction errors even in regions near domain boundaries.
• The model maintains stable rollouts over multiple timesteps, despite being trained only on next-step predictions, indicating robust temporal generalization.
• The GCN surrogate provides a 220-order-of-magnitude speedup compared to high-fidelity numerical simulators, enabling rapid uncertainty quantification in subsurface applications.
• The inclusion of 5 additional PEBI mesh samples in training improved prediction accuracy on unstructured grids, confirming the value of domain-informed data augmentation.